Moving from material to texture, how do I keep diffuse/specular settings?
It is possible to make color palette automatically through scripting in both unity and blender, but with given info and case it's beyond the scope of this question. You can create texture manually in blender through setting UV/Image view from view to paint, creating new 2px*2px image, and painting it with brush with radius 1.

For your other question, there is no simple automatic way to put unity-compatible specular data into your texture. The specularity behaviour depends purely of the choice of the shader and supplied textures in unity, and without further info it's hard to say much about how to proceed with that and also it's not really blender-related question, so you need to go look for unity-related tutorials and forums. 


   To get the look in that tutorial you need to switch your material's shader in Unity from standard to Legacy > Diffuse